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Almost periodic solutions and asymptotic stability
JOURNAL
OF MATHEMATICAL
Almost
ANALYSIS
Periodic
AND
2 1, 136-149
APPLICATIONS
Solutions
and Asymptotic
(1968)
Stability
GEORGE SEIFERT* Iowa
State
University,
Submitted
Ames, Iowa
50010
by J. P. LaSalle
INTRODUCTION
In a recent paper [l], G. R. Sell has introduced a type of stability he refers to asuniform asymptotic stability, but which is in generalweaker than what is usually referred to by that terminology; cf., for example, [2]. He obtains, in terms of this weaker stability, sufficient conditions that a periodic system of ordinary differential equations have a periodic solution. In this paper we show that for systemswith almost periodic time dependence, if the zero solution is uniformly asymptotically stable in the senseof Sell, then it is also in the usual sense.More generally, we show that if every solution whosevalues remain in a given compact set is uniformly asymptotically stable in the senseof Sell, then each such solution is uniformly asymptotically stable in the usual sense. We also show that a similarly weaker type of uniform asymptotic stability in the large, alsointroduced in [1], doesnot imply the type of stability usually referred to by that terminology, even for almost periodic systems.However, such a weaker type of asymptotic stability in the large doesyield an existence theorem for almost periodic solutions of almost periodic systemsfrom which Sell’s Theorem 5 in [l] follows asa corollary.
1.
NOTATION,
DEFINITIONS,
AND
EXAMPLES
We consider first a systemof differential equationsgiven by x’ = F(t, x),
(1)
where x and F(t, x) are elementsof R”, t is in R, F is continuous on R x R”, and F(t, 0) = 0 for all t in R; here R* denotes Euclidean n-spaceover the reals, Ii1 = R, and for x in Rn, we denote by 1x ] the Euclidean norm of x. * This research was supported by the National Science Foundation GP-58898.
136
under Grant
ALMOST
We Clearly The is said
PERIODIC
SOLUTIONS
137
denote by x(t, t, , x,,) a solution of (1) such that ~(t, , t,, , x0) = q, x = 0 is a solution of (1). following definitions are standard; cf. [2], [3]. The solution x = 0 to be:
(1.1) uniformly stable (abbreviated U.S.) if given .z > 0, there exists a 6(r) > 0 such that j x0 / < S(E) implies / x(t, t, , q,) 1 < F for all t 2 t, , and t, in R. (1.2) uniformly asymptotically stable (abbreviated u.a.s.), if it is uniformly stable, and if there exists a r > 0 such that for each E > 0 there exists a T(E) > 0 such that j x,, j < Y implies / x(t, t, , x,,) j i E for t > t, + T(E) and t, in R. (1.3) uniformly asymptotically stable in the large (abbreviated u.a.s.1.) if it is uniformly stable and if for each B > 0 and x0 in R” there exist positive numbers T(E, x,,) and B(x,) such that ( x(t, t, , x,,) / < E for t > t, + T(E, .q,), and / x(t, t, , x,,) 1 < B(x,) for t > t, , t, in R. The following definitions appear in [l]; we distinguish them from the preceding with the prefix “weakly.” The solution x = 0 is said to be (1.4) weakly uniformly asymptotically stable (abbreviated w.u.a.s.) if it is uniformly stable and if there exists a r > 0 such that 1 x,, j < Y implies (t, t, , x0)---f 0 as t + + 00. (1.5) weakly unaformly w.u.a.s.1.) if it is uniformly
asymptotically stable in the large (abbreviated stable and if for each x,, in Rn, x(t, t, , x0) + 0
ast++m.
We also use the following
definition:
(1.6) a function x(t) on R to Rn is bounded on R if there exists a B > 0 such that i x(t) / < B for all t in R. REMARK 1. In required that t, 3 and conditions on our definitions for
most standard definitions of uniform stability, it is only 0 hold. Since we are interested in almost periodic systems solutions which are bounded on R, we have formulated all t, in R.
REMARK 2. It follows easily that if x = 0 is u.a.s.1. then it is u.a.s., and if it is w.u.a.s.l., it is w.u.a.s. It follows also that if F is independent of or periodic in t, then the weak stabilities imply the corresponding strong types for the zero solution of (1). REMARK 3. An example of a system where the zero solution but not u.a.s. is the case where x and F(t, X) are scalars and
F(t, x) = - 2tx
for
t >, 0,
0
for
t < 0.
is w.u.a.s.
138
SEIFEXT
In this caseif to > 0, x(t, to , x0) = xoe-(ta-to*) xOeto2
for
t a 0,
for
t < 0,
while if to < 0, for
t < 0,
for
t > 0.
44 to , x0) = x0 xoe- @
In this casethe zero solution is in fact w.u.a.s.1.We also note that each solution is bounded on R. REMARK4. An example of a system where the zero solution is w.u.a.s. but not totally stable; i.e., stable under constantly acting disturbances,(cf. [2] or [3] for the definitions), is given by (1) where F(t, X) is a scalardefined by F(t, x) = 0,
t G 0,
- 2tx,
o
-25
t>
1;
we omit the details. This is of interest here since the sufficient conditions for the existenceof an almost periodic solution for an almost periodic system due to R. K. Miller in [4] can actually be given in terms of total stability instead of in terms of uniform asymptotic stability, which implies total stability; cf. [3]. REMARK 5. The following example is of a scalar equation with almost periodic time dependencesuch that the zero solution is w.u.a.s.1.but is not u.a.s.1. Define
qt, x) = - x
for
O
- 1 + (1 - 2fO)) (x - 1)
for
1
- xf(t)
for
x > 2,
andF(t, x) = - F(t, - x) for x < 0. Heref (t) is the almost periodic function constructed by Miller and Conley in [9] which has the properties that (i) (ii) that
Jif (t) dt -+ + co as T-+ + co, and there exist real sequences{t,}, (T,}, each -+ + co as n + co, such ha+=, f (t> dt < s t”
- n
for
n = 1, 2,... .
139
ALMOST PERIODIC SOLUTIONS
The zero solution is clearly u.a.s. That it is not u.a.s.1.follows easily from (ii), and that it is w.u.a.s.1. is alsonot difficult to verify; we omit the details. We consider now the system x’ =f(t,
4,
(2)
where f has the sameproperties which F has in (1) except that we do not require f (t, 0) = 0. Let x = p(t) be a solution of (2) defined for all t in R. Then we define each type of stability (1.l)-(1.5) for 9)as follows: the solution e) of (2) has the particular stability defined in (1 . j), j = 1, 2, 3, 4, 5, if the zero solution
of (3)
x’ = f(t, x + v,(t)> - f(t, v(t)) has that type of stability.
2.
ALMOST
PERIODIC SYSTEMS AND THEIR STABILITY
We say that a function
PROPERTIES
f of (t, X) is almost periodic in t uniformly for x in
R* (abbreviated a.p.u. (Rn)) if f is continuous at each (t, x) in R
x
Rn and if
f is almost periodic in t uniformly for x in each compact subsetof Rn; cf. [5] for the definition of this last concept. It is well know that if f is a.p.u. (Rn), then (iii) for each real sequence{tn} there exists a subsequence(t;} and a function g which is a.p.u. (Rn) such that f (t + t; , x) -+g(t, x) as k - co, the limit being uniform for t in R and x in any compact subset of R”; and (iv)
there exist real sequences{t,} and {th} such that if n + 00, t, -+ + on, - co, f(t + tn, x)-+f(t,x), and f(t + tk, x)-+f(t,x), each limit being uniform for t in R and x in any compact subsetof R”.
tL--t
If f is a.p.u. (Rn) and p is a solution of (2) bounded on R, x + y(t)) is not necessarilya.p.u. (Rn). Thus the corresponding Eq. (3) cannot in this casebe consideredan a.p. system even though (2) is. For eachf which is a.p.u. (R”), we denote by H(f) the set of all functions g such thatf(t + t, , X) +g(t, x) as k + co for some real sequence{tk}, the limit being uniform for t in R and x in any compact subsetof R”. It follows easily that if g is in H( f ), then REMARK 6.
f(t,
(v) f is in H(g), and in factg(t - t, , x) +f(t, {tk} and the uniform approach being as above;
X) ask --f co, the sequence
(vi) there exists a real sequence{&}, i, --t + co as n ---t cg, such that f(t + ik , x) +g(t, x) as k + 03 uniformly for t in R and x in any compact subset of An.
140
SEIFERT
We state without proof the following case of a result due to Kamke [6]:
lemma which is in a sense a special
LEMMA 1. Suppose that g, , k = 1, 2,..., are functions continuous on R x R” to Iin and that g, -+g as k -+ co unsformly on each compact subset of R x R”. Then if c&t) is a solution of x’ = gk(t, x) such that ~(0) -+ x0 , as k---f co, and 1 vk(t) 1 < B for all t in I, a real interval containing t = 0, and k = 1, 2,..., then there exists a subsequence {ki} of the sequence {k} of integers, and a solution g, of x’ = g(t, x) such that $t) --f v(t) as j + co, the limit being uniform on compact subsets of I, and such that ~(0) = x0 and I y(t) 1 < B for t in I. THEOREM 1. Let f be a.p.u. (R”) and suppose there exists a B > 0 such that each solution CPof (2) satisfying / v(t) 1 < B for all t in R is w.u.a.s. Then each such solution v is u.a.s.
In the proof of this theorem we need the following LEMMA equation
2.
Let f and v satisfy the hypotheses of Theorem 1, and consider the Y’ =f
(6 Y + v(t)) -f
(4 v(t))*
(3V)
Then there exists a number r0 > 0 independent of 9 such that if 1yo 1 < r,, , then y(t, t, , y0 , 9)) + 0 as t + 00; here y(t, to , y,, , q) is a solution of (3~) such such that r(to , to , y. , P’) = y. . PROOF. Let v be a solution of (2) such that I p)(t) I < B for all t in R and let {r(p))} denote the set of all r > 0 such that y(t, to , y. , p’) --t 0 as t -+ CO whenever / y. 1 < r. By hypothesis, {r(y)} is nonempty. Define
P(F) = min (sup NV),)>,11 and r. = inf {p(v)}, the infimum being taken over the set of solutionsg, of (2) satisfying the hypothesesof the theorem. Supposer. = 0. Then there exists a sequence{v~} of such solutions such that p(vk) + 0 and ~~(0) --+x0 as k + co. Clearly 1x0 1< B. By applying Lemma 1 with glc(t, x) -f(t, x), k = 1,2,..., we find that for a suitable subsequence,which we again index by k, vK(t) --f To(t) as k -+ co uniformly on compact subsetsof R; here v. is a solution of (2) such that I To(t) I < B, and is therefore w.u.a.s. Hence there exists a r(& > 0 in the set {r(vo)} defined above. For a fixed to in R there exists an integer k so large that
I mdto>- To(tu>I < ry and p(yk)
I,/.
(3.1)
ALMOST
From the definition
PERIODIC
of r((ps) it follows
141
SOLUTIONS
that
I CD&)- vo(t>I --t 0
t-+c.o.
as
(3.2)
Also if x0 satisfies
r(vJo) I x0 - %(to) I -c 2 9 it follows
that / x0 - yo(f,) 1 < r(&.
Consequently
I x(t, to , x0) - 90(t) I + 0 It follows,
using this last limit
(3.3)
as
t+
co.
as
t-P
co.
and (3.2), that
I 46 to , x0) - a(t)
I -+ 0
But then, using (3.3), we must have sup {TV)) >, r(q,)/2, and hence p(cpk) > min {r(p,)/2, l}. This contradicts (3.1) and we therefore must have y. > 0. Clearly this r. satisfies the required conditions, and the lemma is proved. PROOF OF THEOREM 1. Consider a solution v of (2) satisfying / q(t) < B for all t in R. We must prove that the zero solution of the corresponding equation (397) is u.a.s. Since this zero solution of (39) is w.u.a.s., it is U.S., and hence, there exists a 6(r,) > 0 such that I y. 1 s: 6(r,) implies
for
I Y(C to ! Yo) I < f-0
t > to,
to in
R;
here r. is the number given in Lemma 2, and y(t, to , yo) is a solution of (391). Now let E > 0 be given, and fix y. such that / y. j < S(yo). We will show that there exists a T(E) > 0 such that for each to in R, there exists t, , to f t, < to + T(E), such that ( y(tl , to , yo) / < B(E), where 6(c) is as determined by the uniform stability of y = 0. It will then clearly follow that 1y(t, to, yo) I < E for t > to + T(E), which is what is to be proved. Suppose no such T(E) exists. Then for each integer n 2 1, there exists a t, such that I Y(h tn YYo) I 2 S(E)
for
t, < t < tn + n.
(4)
We shall show that there exists a subsequence {Q) of the sequence of integers and a function G such that F(t + tnk , y) -+ G(t, y) as k + a uniformly on compact subsets of R x Rn; here F(t,y) =f(s,y -t v(t)) -f(t, p)(t)). Since / p)(t) j < B for t in R, and since f is a.p.u. (R”), it follows using (iii) and Lemma 1, that for a suitable subsequence (ah} of the integers, x) +g(t, X) and p(t + tn,) + t,b(t) as k + co, the convergence f(t + tnl,> being uniform on compact subsets of R x R” and R respectively. Here 4 is a solution of
x’ = g(t, x)
SEIFERT
142
such that / #(t) < B for t in R. If we define Fk(t, y) = F(t + tQk, y) and G(t, y) = g(t, y + 4(t)) - g(t, #(t)), it follows that Fk(t, y) - G(t, Y) as k -+ co uniformly on compact subsetsof R x R”; we omit the details. If we define t: = t % ’ and yk(t) = y(t + t$, tj,!, ys), then clearly yk(t) is a solution of y’ = F,(t, y) such that ~~(0) = y,, . Since 1y,, 1 < 6(r,), it follows that 1yk(t) 1 < r0 for t > 0, K = 1, 2,... . Thus by Lemma 1, there exists a subsequence, which we again index by K, such that yk(t) + z(t) as k -+ CO uniformly on compact subsetsof R+, where z is a solution of y’ = G(t, y) such that z(0) = y,, , and R+ is the set of t 2 0. Clearly for fixed t > 0, there exists a k sufficiently large so that
I @> I 3 Iyr(t> I - IyIm - z(t) I > y
(5)
where we have also used (4). Also since 1yk(t) 1 < r0 for t > 0 and k = 1, 2,..., it follows that
I z(t) I < y.
for
5>
0.
(6)
Now since g is in H(f), it follows that f is in H(g), and there exists a sequence (~~1, rk -+ + co as k + CO such that g(t + rk , x) -f (t, x) as k---f co uniformly for t in R and x in any compact subsetof R”. By use of Lemma 1 again, there exists a subsequenceof {T%},which we again denote by (Q}, such that #(t + Q) +$(t) as R+ co uniformly on compact subsetsof R; here 8 is a solution of (2) such that I p(t) I < B for t in R. If we define et,
Y) =f
(t9 Y + G(t)) -f
(6 F(t)),
it clearly follows that G(t + Tk , y) +F(t, y) as k + co, uniformly on compact subsetsof R x Rn. Clearly for each integer k, zk(t) = z(t + Q) is a solution of y’ = G(t + 7k , y), and by use of Lemma 1 again, there exists a subsequenceof {TV}, which we again denote by {Q}, such that zk(t) +9(t) as k + co uniformly on compact subsetsof R+; here y(t) is a solution of y’ = p(t, y). Now for fixed t > 0, there exists a R so large that
19(t) I z I +(t) I - I %(t) -7(t) I a *q where we have used (5) and the fact that 7k > 0 for K sufficiently large. Also from (6) and this samefact, it follows that Iy”(0) I < r, . But this implies y(t) + 0 as t + + co, which contradicts (7). This proves the theorem. COROLLARY 1. If f is a.p.u. (Rn), and x = 0 is a solution w.u.a.s., then it is u.a.s.
of (2) which is
ALMOST PROOF.
This corollary follows
PERIODIC
143
SOLUTIONS
immediately
from Theorem
1 by taking
B -0.
We observe here that Miller and Conley have given an example of a linear scalarsystem in which f is a.p.u. (P), the zero solution is asymptotically stable, but not uniformly stable; cf. [9]. 2. Let the hypotheses of Theorem I be saiisfied and in addition, suppose for each g in H(f) there exists a constant L(g) such that THEOREM
I ‘dc 4 - g(4 Y) f L(g) I A- -Y
I
for 1 x ) < B, 1y 1 < B, and 5 in R. Then each so&ion 1 p)(t) 1 < B for aZZ t in R is a.p.
g, of (2) such Ihat
PROOF. We first show that eachsolution ‘pof (2) satisfying the hypotheses of this theorem is separated(in the senseof Amerio [lo]) in the set S(B) of all x in Rn such that 1x I < B; i.e., for eachsuch v we show that there exists p(g)) > 0 such that for any solution +!J of (2) distinct from y such that / #(t)
I F(t) - (Ir(t>I 2 P(cp> for all
tER.
Since by Theorem 1 eachsuch 9 is u.a.s., each such ‘p is totally stable (i.e., stable under constantly acting disturbances) [3]. Supposesomesuch 9” = p1 is not separatedin S(B). Since ~‘1is u.a.s., there exists 6, > 0 such that if to is in R, then as 2+ co, I x(t, to ?x0) - dt) I - 0 provided 1x0 - q(to) / < 6,. But if v1 is not separatedin S(B), there exists a solution 9+ # p1 such that / pJt) 1< B for all t in R and 1yz(to) -- cpl(to) 1 < Sofor someto in R. Hence / cpz(t) - yl(t) / -+ 0 ast -+ co, and it follows also from the facts that v1 is u.a.s. and v1 # ~~ , that there exists a 6, > 0 such that 1q+(t) - VI(t) 1> 8, for t < to. Let {tk} be a sequencesuch that as K + 03, we have t, -+ - 00, f (t + t, , x) -f (t, cc), and Fj(t + tk) --+ a,+(t), j = 1,2; here the #j are solutions of (2), and the convergence is uniform on compact subsetsof R x R* and R, respectively (cf. Lemma 1). It follows easily that ( &(t) - $,(Q 2 6, for each t in R; we omit the details. Now pj(t + tk) is a solution of
x’ =f(c 4 + (f (t + h, 4 -f (C x)),
(4.1)
and #j is a solution of (2). Hence it follows from the total stability of #j that there exists a N(6,) > 0 such that for K > N(S,), we have for
t 20,
j=
1,2.
144
SEIFERT
For each such K and t > 0, it follows
easily that
and since also 1v,(t + tk) - v,(t + tk) ( + 0 as t -+ CO, we have for t sufficiently large that j #r(t) - #s(t) / < 6, , a contradiction. We conclude that yI is separated in S(B). We next show that if g is in H(f), then each solution # of x’ = g(t, x) Gw such that 1#(t) j < B f or all t in R is separated in S(B). To this end, we first observe that the set of such solutions of (2g) is necessarily finite. Let us denote the set of such solutions of (2) by {vj},j = 1,2,..., m. There exists a sequence {tk}, k = 1, 2,..., such that as k --f co, t, -+ + co, f(t + tk , X) -+g(t, x) (cf. (v)), and qj(t + tic) + #j(t) for j = 1, 2,..., m; here the convergence is uniform on compact subsets of R x Rn and R, respectively, and #j are solution of (2g). Since each p)j , j = I,2 ,..., m, is u.a.s., it follows that each #j , j = 1, 2,..., m, is a u.a.s. solution of (2g); cf. Lemma 4 in [7]. Now let # be a solution of (2g) such that 1z/(t) / < B for all t in R and such that 4 # #j , j = 1, 2,..., m. By using a suitable subsequence of {tk}, which we again denote by {tk}, it follows that #(t - tk) + Fdt) for some j, 1 < j < m, uniformly because g(t - t, , x) +f(t, x) as because there are no solutions of (2) the vi . However, there also exists using a suitable subsequence of (tk},
as
R+co
(4.2)
on compact subsets of R; this is true K -+ co uniformly on R x S(B), and with values contained in S(B) other than an integer i, 1 < i ,< m, such that by which we again denote by {tk}, we have
(bdt - tk> -+ 944
as
k-+co
(4.3)
uniformly on compact subsets of R. For if this were not true, there would exist integers p, 4, and r, p # 4, r fj, such that &,(t - tJ -+ ~)r(t) and &(t - tk) ---f y+(t) as k -+ co uniformly on compact subsets of R, where again a suitable subsequence, denoted by (tk}, is involved. But then 1#,( - tk) - z+,(- tk) ) -+ 0 as K --f co, and since tk -+ + cc as k + CO, and 9, # &, , it would follow that #, cannot be a.u.s. solution of (2g). From (4.2) and (4.3) it follows that / #( - tk) - Q&(- tk) + 0 as k -+ co, and since &. is a U.S. solution of (2g), we conclude that # = +t Yence the set of solutions with values in S(B) for all t in R consists of the fim set {I#~}, j = 1, 2,..., m. By using the same argument by which we proved ea I solution vi of (2) was separated in S(B), we find that each tij , j = 1, 2,..., 1, is separated in S(B) as a solution of (2g).
ALMOST
PERIODIC
145
SOLUTIONS
We finally apply a result due to Amerio [lo] and the theorem is proved. We turn now to some results involving certain types of asymptotic stability in the large. LEMMA 3. Letfbe a.p.u. (Rn), and suppose v is a solution of (2) bounded on R and w.u.a.s.1. Then 9 is the only solution of (2) bounded on R. PROOF. Suppose Q!Iis another solution of (2) bounded on R. Since f is a.p.u. (Rn), there exists a sequence {tr}, t, ---f - co as k-t co, such that f (t + t, , X) -f (t, X) uniformly for t in R and x in any compact subset of R”; cf. (iv). Using Lemma 1, it follows that there exists a subsequenceof {tk) which we again denote by {tk} such that p)(t + tr) -+ q(t) and #(t + tk) - $ (t) as k + co uniformly on compact subsetsof R; here q and 4 are solutions of (2). Thus G(t) - p)(t) -+ 0 and $(t) - q(t) -+ 0 as t + + 00, and it follows that
$5(t) - $(t) --+ 0
as
t--t+00
(8)
Suppose there exists a to in R such that 1I - $(t,) 1 = l 0 > 0. It follows, since v is U.S.,that there exists a y0 > 0 such that for
I 40 - 6(t) I 2 ro
t < to 7
(9)
and also,by (8) that there exists tl in R such that
(10) Clearly for all k sufficiently large, we have t, + t, < to , and hence, using (9), I +(tl + tk) - v(t, + tk) I > ro. It follows that I J(h) - $(tl) I 3 y. which contradicts (10). Thus q(t) = $(t) for all t in R and our proof is complete. REMARK 7. It is easily seenthat the conclusion of Lemma 3 will hold if the hypothesis that f is a.p.u. (Rn) is replaced by an obviously weaker one; namely, that there exists a sequence{td, tk + - cc as k -+ 00, such that f (t + tk , x) -+f (t, x) as k -+ co, uniformly on compact subsetsof R x Rn.
LEMMA 4. Let the hypotheses of Lemma 3 be satisfied suppose that for each g in H( f ), the initial value problem for x’ =g(t, has a unique solution. Then solution bounded on R. PROOF.
f (t + c&21
t,
I-IO
for
x)
euch equation
and in addition,
GM (2g) there exists one and only one
Since g is in H(f), there exists a sequence {tk} such that , x) +g(t, x) as k -+ 00 uniformly for t in R and x in any compact
146
SEIFERT
subsetof Rn. It is no lossof generality to supposealsothat CJJ(~ + tJ -+ #r(t) as k -+ COuniformly on compact subsetsof R; hence t,+(t) is a solution of (2g) bounded on R. Let #a be another solution of (2g) bounded on R. Then since g(t - t, , x) +f(t, x) as k + cc uniformly for t in R and x in any compact subset of Rn, we have, by use of an appropriate subsequenceif necessary, which we again denote by {tk), that w
- tk) + W)
and
$z(t - tk) + f&(t)
as
k -+ co
uniformly on compact subsetsof R; here @jrand +a are bounded solutions of (2). By Lemma 3, ~$5~ = & = v, and it follows that ) #l(t - tk) - &(t - tk) / + 0
as
K + 0~)
(11)
uniformly for t in compact subsetsof R. We now show that h(t) - y&(t) -+ 0 as t--j + co. To this end, we assert that under the conditions of our lemma, the solution #r of (2g) is U.S.; for a proof of this, cf. Lemma 4 in [7]. Thus given E > 0, there exists a 8(e) > 0 as specified by the uniform stability of #r , and consequently alsoa k = K(c) such that I A(- tk) - Atk) I < %4; this last estimatefollows clearly from (11). It follows that I VW) - #z(t) I < e
for
t>
--k,
and therefore we have shown that a+&(t)- q&(t) + 0 as t --+ CO,and conclude that #r is a solution of (2g) which is w.u.a.s.1. But then, by Lemma 3 as applied to (2g) and +I , it follows that #r = #a , and the proof is complete. REMARK 8. It may be of interest to observe that the condition on the initial value problem for systems (2g) in the hypotheses of Lemma 3 is needed only to conclude, by Lemma 4 in [7], that the solution #r of (2g) is U.S.if its “prototype,” p is. Let f be a.p.u. (Rn). It is known that corresponding to such an f there exists a unique sequence{A,} of real numbers, the so-calledFourier exponents off. We call the set of all numbers of the form n,X, + nab, + *a*+ nkhk, where KZ~ ,..., nb are integers, the module off. THEOREM 3. Let f be a.p.u. (Rn) and suppose for each g in H(f) the initial value problems for (2g) have unique solutions. Let q~ be a solution of (2) which is w.u.a.s.1. and bounded on R. Then v is the only solutions of (2) which is bounded on R, is a.p., and its module is contained in the module off. PROOF.
The fact that y is a.p. follows immediately from Lemma 4 and
ALMOST
PERIODIC
SOLUTIONS
147
a result due to Amerio [lo]. The fact that the module of 9 is contained in the module off follows from the fact that for each g in H(f), (2g) has exactly one solution bounded on R and that therefore the argument Favard [8] uses for linear systemscan be used here; we omit the details. It may be of interest to observe that this Theorem apparently does not follow from Lemma 3 and Theorem 2. In the first place we do not assumea Lipschitz condition here such as was required in Theorem 2, and we also do not get the information on the module of the a.p. solution from Theorem 2. COROLLARY 2. Let f be periodic in t with period w > 0, and suppose the initial value problems for (2) have unique solutions. Then if F is a solution of (2) which is w.u.a.s.1. and is bounded on R, it is periodic with period W. PROOF. If the initial value problems for (2) have unique solutions, it follows easily that the initial value problems for eachsystemx’ = f (t + h, x), where h is a constant, 0 < h < w, also have unique solutions. But in this case H(f) consists of all functions of the form f (t -\ h, x), 0 < h < O. Hence the hypothesesin Theorem 2 are satisfied. Since the module off is now (2~ k/w}, k = 0, & 1, f 2,..., the Fourier exponents of 91must be integer multiples of 2rr/w. From this it follows easily that C,Jis periodic with period W, and proof of the corollary is complete. This corollary sharpensSell’s Theorem 5 in [I] which assertsthe existence of a periodic solution of period RWfor someinteger k. The existence in this caseof a solution of period w hasalsobeen establishedby T. Yoshizawa using other methods. REMARK 9. Some apparently open questions suggest themselves. First, if f is a.p.u. (R”), can the hypothesis in Theorem 1 that all solutionsg, of (2) such that / y(t) 1< B for all t in R be w.u.a.s. be replacedby the condition that at least one such v has this stability property ? Also iff is periodic in t with period w the question may arise asto whether the existenceof a w.u.a.s. bounded solution implies the existenceof a periodic solution of period w. Sell’s Theorem 4 in [l] assertsonly the existence of a periodic solution of period KU, k an integer. The following example provides a negative answer to this last question. Define cos t sin t 44 = (- sin t cost)
consider the system:
E; = - (1 56= Ed1 - Cf22).
148
SEIFERT
If E = (fl , ta), and f(f) = (- 4, , (a(1 formed by x = A(# into the system: x’ =Fyt,
Es2)), this
system
x)
is trans-
(12)
where qt,
x) = A’(t)
AT(t) x + A(t)f(AT(t)
x),
and
A-l(t) = AT(t) = (;g : - zb”,:) . By using the facts that f(E) = -f(f), A(t + CT) = - A(t), A’(t + CT) = - A’(t), and AT(t + n) = - AT(t), we observe that F(t + ?r, X) = F(t, X) for all (t, X) in R x R2. Now consider the 3-dimensional system x’ = F( t, x)
4 = Ax, 4,
(13)
where
1 - 2 I x 12,
I x I e +2;
here x = (xi , x2) and I x 1 = (xl2 + x2a)i/a. It follows that (13) is in t with period V, and that (- sin t, cos t, 0) and (sin t, - cos I, 0) tions of (13) of p eriod 27. It is easy to show that these are the only solutions of (13) and that they are u.a.s.; we omit the details. We also that any solution (x(t), xs(t)) of (13) such that
periodic are soluperiodic observe
(x(&J, x2(&)) = (u cos to , - a sin t, , c) where a and c are constants becomes unbounded as t + co, while any other solution tends to one of the two periodic solutions as t + CO. This example, and in particular, this last observation suggests the apparently open question: if each solution of a periodic system is bounded for t > 0 and if each solution bounded for all t is u.a.s., then is there among such bounded solutions a periodic solution with the same period as the system ? REMARK 10. It is clear that our hypotheses on the boundedness of solutions could have been weakened in the sense that any condition of the form “1 y(t) 1 < for all tin R” could be replaced by “a(t) in K, a compact subset of Rn, for all t in Rn”.
ALMOST
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149
REFERENCES 1. G. R. SELL. Periodic solutions and asymptotic stability. J. Diff. Eqs. 2 (1966), 143-157. 2. T. YOSHIZAWA. Stability theory by Liapunov’s second method. Math. Sot. Japan (1966). 3. J. MASSERA. Contributions to stability theory. Ann. Math. 64 (1956), 182-206. 4. R. K. MILLER. Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions. /. D$f. Eqs. 1 (1965), 337-345. 5. J. K. HALE. “Oscillations in Nonlinear Systems,” McGraw-Hill, New York, 1963. 6. E. KAMKE. Zur Theorie der Systeme gewijhnlicher Differentialgleichungen. II Acta Math. 58 (1932), 57-85. 7. G. SEIFERT. A condition for almost periodicity with some applications to functionaldifferential equations. J. Diff. Ep. 1 (1965), 393-408. 8. J. FAVARD. “Lecons sur les Fonctions Presque-Periodiques,” pp. 88-91. GauthierVillars, Paris, 1933. 9. C. C. CONLEY AND R. K. MILLER. “Asymptotic stability without uniform stability: almost periodic coefficients,” J. Diff. Eqs. 1 (1965), 333-336. 10. L. AMERIO. Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitati. Ann. Mat. Pura Appl. 34 (1955), 97-119.
OF MATHEMATICAL
Almost
ANALYSIS
Periodic
AND
2 1, 136-149
APPLICATIONS
Solutions
and Asymptotic
(1968)
Stability
GEORGE SEIFERT* Iowa
State
University,
Submitted
Ames, Iowa
50010
by J. P. LaSalle
INTRODUCTION
In a recent paper [l], G. R. Sell has introduced a type of stability he refers to asuniform asymptotic stability, but which is in generalweaker than what is usually referred to by that terminology; cf., for example, [2]. He obtains, in terms of this weaker stability, sufficient conditions that a periodic system of ordinary differential equations have a periodic solution. In this paper we show that for systemswith almost periodic time dependence, if the zero solution is uniformly asymptotically stable in the senseof Sell, then it is also in the usual sense.More generally, we show that if every solution whosevalues remain in a given compact set is uniformly asymptotically stable in the senseof Sell, then each such solution is uniformly asymptotically stable in the usual sense. We also show that a similarly weaker type of uniform asymptotic stability in the large, alsointroduced in [1], doesnot imply the type of stability usually referred to by that terminology, even for almost periodic systems.However, such a weaker type of asymptotic stability in the large doesyield an existence theorem for almost periodic solutions of almost periodic systemsfrom which Sell’s Theorem 5 in [l] follows asa corollary.
1.
NOTATION,
DEFINITIONS,
AND
EXAMPLES
We consider first a systemof differential equationsgiven by x’ = F(t, x),
(1)
where x and F(t, x) are elementsof R”, t is in R, F is continuous on R x R”, and F(t, 0) = 0 for all t in R; here R* denotes Euclidean n-spaceover the reals, Ii1 = R, and for x in Rn, we denote by 1x ] the Euclidean norm of x. * This research was supported by the National Science Foundation GP-58898.
136
under Grant
ALMOST
We Clearly The is said
PERIODIC
SOLUTIONS
137
denote by x(t, t, , x,,) a solution of (1) such that ~(t, , t,, , x0) = q, x = 0 is a solution of (1). following definitions are standard; cf. [2], [3]. The solution x = 0 to be:
(1.1) uniformly stable (abbreviated U.S.) if given .z > 0, there exists a 6(r) > 0 such that j x0 / < S(E) implies / x(t, t, , q,) 1 < F for all t 2 t, , and t, in R. (1.2) uniformly asymptotically stable (abbreviated u.a.s.), if it is uniformly stable, and if there exists a r > 0 such that for each E > 0 there exists a T(E) > 0 such that j x,, j < Y implies / x(t, t, , x,,) j i E for t > t, + T(E) and t, in R. (1.3) uniformly asymptotically stable in the large (abbreviated u.a.s.1.) if it is uniformly stable and if for each B > 0 and x0 in R” there exist positive numbers T(E, x,,) and B(x,) such that ( x(t, t, , x,,) / < E for t > t, + T(E, .q,), and / x(t, t, , x,,) 1 < B(x,) for t > t, , t, in R. The following definitions appear in [l]; we distinguish them from the preceding with the prefix “weakly.” The solution x = 0 is said to be (1.4) weakly uniformly asymptotically stable (abbreviated w.u.a.s.) if it is uniformly stable and if there exists a r > 0 such that 1 x,, j < Y implies (t, t, , x0)---f 0 as t + + 00. (1.5) weakly unaformly w.u.a.s.1.) if it is uniformly
asymptotically stable in the large (abbreviated stable and if for each x,, in Rn, x(t, t, , x0) + 0
ast++m.
We also use the following
definition:
(1.6) a function x(t) on R to Rn is bounded on R if there exists a B > 0 such that i x(t) / < B for all t in R. REMARK 1. In required that t, 3 and conditions on our definitions for
most standard definitions of uniform stability, it is only 0 hold. Since we are interested in almost periodic systems solutions which are bounded on R, we have formulated all t, in R.
REMARK 2. It follows easily that if x = 0 is u.a.s.1. then it is u.a.s., and if it is w.u.a.s.l., it is w.u.a.s. It follows also that if F is independent of or periodic in t, then the weak stabilities imply the corresponding strong types for the zero solution of (1). REMARK 3. An example of a system where the zero solution but not u.a.s. is the case where x and F(t, X) are scalars and
F(t, x) = - 2tx
for
t >, 0,
0
for
t < 0.
is w.u.a.s.
138
SEIFEXT
In this caseif to > 0, x(t, to , x0) = xoe-(ta-to*) xOeto2
for
t a 0,
for
t < 0,
while if to < 0, for
t < 0,
for
t > 0.
44 to , x0) = x0 xoe- @
In this casethe zero solution is in fact w.u.a.s.1.We also note that each solution is bounded on R. REMARK4. An example of a system where the zero solution is w.u.a.s. but not totally stable; i.e., stable under constantly acting disturbances,(cf. [2] or [3] for the definitions), is given by (1) where F(t, X) is a scalardefined by F(t, x) = 0,
t G 0,
- 2tx,
o
-25
t>
1;
we omit the details. This is of interest here since the sufficient conditions for the existenceof an almost periodic solution for an almost periodic system due to R. K. Miller in [4] can actually be given in terms of total stability instead of in terms of uniform asymptotic stability, which implies total stability; cf. [3]. REMARK 5. The following example is of a scalar equation with almost periodic time dependencesuch that the zero solution is w.u.a.s.1.but is not u.a.s.1. Define
qt, x) = - x
for
O
- 1 + (1 - 2fO)) (x - 1)
for
1
- xf(t)
for
x > 2,
andF(t, x) = - F(t, - x) for x < 0. Heref (t) is the almost periodic function constructed by Miller and Conley in [9] which has the properties that (i) (ii) that
Jif (t) dt -+ + co as T-+ + co, and there exist real sequences{t,}, (T,}, each -+ + co as n + co, such ha+=, f (t> dt < s t”
- n
for
n = 1, 2,... .
139
ALMOST PERIODIC SOLUTIONS
The zero solution is clearly u.a.s. That it is not u.a.s.1.follows easily from (ii), and that it is w.u.a.s.1. is alsonot difficult to verify; we omit the details. We consider now the system x’ =f(t,
4,
(2)
where f has the sameproperties which F has in (1) except that we do not require f (t, 0) = 0. Let x = p(t) be a solution of (2) defined for all t in R. Then we define each type of stability (1.l)-(1.5) for 9)as follows: the solution e) of (2) has the particular stability defined in (1 . j), j = 1, 2, 3, 4, 5, if the zero solution
of (3)
x’ = f(t, x + v,(t)> - f(t, v(t)) has that type of stability.
2.
ALMOST
PERIODIC SYSTEMS AND THEIR STABILITY
We say that a function
PROPERTIES
f of (t, X) is almost periodic in t uniformly for x in
R* (abbreviated a.p.u. (Rn)) if f is continuous at each (t, x) in R
x
Rn and if
f is almost periodic in t uniformly for x in each compact subsetof Rn; cf. [5] for the definition of this last concept. It is well know that if f is a.p.u. (Rn), then (iii) for each real sequence{tn} there exists a subsequence(t;} and a function g which is a.p.u. (Rn) such that f (t + t; , x) -+g(t, x) as k - co, the limit being uniform for t in R and x in any compact subset of R”; and (iv)
there exist real sequences{t,} and {th} such that if n + 00, t, -+ + on, - co, f(t + tn, x)-+f(t,x), and f(t + tk, x)-+f(t,x), each limit being uniform for t in R and x in any compact subsetof R”.
tL--t
If f is a.p.u. (Rn) and p is a solution of (2) bounded on R, x + y(t)) is not necessarilya.p.u. (Rn). Thus the corresponding Eq. (3) cannot in this casebe consideredan a.p. system even though (2) is. For eachf which is a.p.u. (R”), we denote by H(f) the set of all functions g such thatf(t + t, , X) +g(t, x) as k + co for some real sequence{tk}, the limit being uniform for t in R and x in any compact subsetof R”. It follows easily that if g is in H( f ), then REMARK 6.
f(t,
(v) f is in H(g), and in factg(t - t, , x) +f(t, {tk} and the uniform approach being as above;
X) ask --f co, the sequence
(vi) there exists a real sequence{&}, i, --t + co as n ---t cg, such that f(t + ik , x) +g(t, x) as k + 03 uniformly for t in R and x in any compact subset of An.
140
SEIFERT
We state without proof the following case of a result due to Kamke [6]:
lemma which is in a sense a special
LEMMA 1. Suppose that g, , k = 1, 2,..., are functions continuous on R x R” to Iin and that g, -+g as k -+ co unsformly on each compact subset of R x R”. Then if c&t) is a solution of x’ = gk(t, x) such that ~(0) -+ x0 , as k---f co, and 1 vk(t) 1 < B for all t in I, a real interval containing t = 0, and k = 1, 2,..., then there exists a subsequence {ki} of the sequence {k} of integers, and a solution g, of x’ = g(t, x) such that $t) --f v(t) as j + co, the limit being uniform on compact subsets of I, and such that ~(0) = x0 and I y(t) 1 < B for t in I. THEOREM 1. Let f be a.p.u. (R”) and suppose there exists a B > 0 such that each solution CPof (2) satisfying / v(t) 1 < B for all t in R is w.u.a.s. Then each such solution v is u.a.s.
In the proof of this theorem we need the following LEMMA equation
2.
Let f and v satisfy the hypotheses of Theorem 1, and consider the Y’ =f
(6 Y + v(t)) -f
(4 v(t))*
(3V)
Then there exists a number r0 > 0 independent of 9 such that if 1yo 1 < r,, , then y(t, t, , y0 , 9)) + 0 as t + 00; here y(t, to , y,, , q) is a solution of (3~) such such that r(to , to , y. , P’) = y. . PROOF. Let v be a solution of (2) such that I p)(t) I < B for all t in R and let {r(p))} denote the set of all r > 0 such that y(t, to , y. , p’) --t 0 as t -+ CO whenever / y. 1 < r. By hypothesis, {r(y)} is nonempty. Define
P(F) = min (sup NV),)>,11 and r. = inf {p(v)}, the infimum being taken over the set of solutionsg, of (2) satisfying the hypothesesof the theorem. Supposer. = 0. Then there exists a sequence{v~} of such solutions such that p(vk) + 0 and ~~(0) --+x0 as k + co. Clearly 1x0 1< B. By applying Lemma 1 with glc(t, x) -f(t, x), k = 1,2,..., we find that for a suitable subsequence,which we again index by k, vK(t) --f To(t) as k -+ co uniformly on compact subsetsof R; here v. is a solution of (2) such that I To(t) I < B, and is therefore w.u.a.s. Hence there exists a r(& > 0 in the set {r(vo)} defined above. For a fixed to in R there exists an integer k so large that
I mdto>- To(tu>I < ry and p(yk)
I,/.
(3.1)
ALMOST
From the definition
PERIODIC
of r((ps) it follows
141
SOLUTIONS
that
I CD&)- vo(t>I --t 0
t-+c.o.
as
(3.2)
Also if x0 satisfies
r(vJo) I x0 - %(to) I -c 2 9 it follows
that / x0 - yo(f,) 1 < r(&.
Consequently
I x(t, to , x0) - 90(t) I + 0 It follows,
using this last limit
(3.3)
as
t+
co.
as
t-P
co.
and (3.2), that
I 46 to , x0) - a(t)
I -+ 0
But then, using (3.3), we must have sup {TV)) >, r(q,)/2, and hence p(cpk) > min {r(p,)/2, l}. This contradicts (3.1) and we therefore must have y. > 0. Clearly this r. satisfies the required conditions, and the lemma is proved. PROOF OF THEOREM 1. Consider a solution v of (2) satisfying / q(t) < B for all t in R. We must prove that the zero solution of the corresponding equation (397) is u.a.s. Since this zero solution of (39) is w.u.a.s., it is U.S., and hence, there exists a 6(r,) > 0 such that I y. 1 s: 6(r,) implies
for
I Y(C to ! Yo) I < f-0
t > to,
to in
R;
here r. is the number given in Lemma 2, and y(t, to , yo) is a solution of (391). Now let E > 0 be given, and fix y. such that / y. j < S(yo). We will show that there exists a T(E) > 0 such that for each to in R, there exists t, , to f t, < to + T(E), such that ( y(tl , to , yo) / < B(E), where 6(c) is as determined by the uniform stability of y = 0. It will then clearly follow that 1y(t, to, yo) I < E for t > to + T(E), which is what is to be proved. Suppose no such T(E) exists. Then for each integer n 2 1, there exists a t, such that I Y(h tn YYo) I 2 S(E)
for
t, < t < tn + n.
(4)
We shall show that there exists a subsequence {Q) of the sequence of integers and a function G such that F(t + tnk , y) -+ G(t, y) as k + a uniformly on compact subsets of R x Rn; here F(t,y) =f(s,y -t v(t)) -f(t, p)(t)). Since / p)(t) j < B for t in R, and since f is a.p.u. (R”), it follows using (iii) and Lemma 1, that for a suitable subsequence (ah} of the integers, x) +g(t, X) and p(t + tn,) + t,b(t) as k + co, the convergence f(t + tnl,> being uniform on compact subsets of R x R” and R respectively. Here 4 is a solution of
x’ = g(t, x)
SEIFERT
142
such that / #(t) < B for t in R. If we define Fk(t, y) = F(t + tQk, y) and G(t, y) = g(t, y + 4(t)) - g(t, #(t)), it follows that Fk(t, y) - G(t, Y) as k -+ co uniformly on compact subsetsof R x R”; we omit the details. If we define t: = t % ’ and yk(t) = y(t + t$, tj,!, ys), then clearly yk(t) is a solution of y’ = F,(t, y) such that ~~(0) = y,, . Since 1y,, 1 < 6(r,), it follows that 1yk(t) 1 < r0 for t > 0, K = 1, 2,... . Thus by Lemma 1, there exists a subsequence, which we again index by K, such that yk(t) + z(t) as k -+ CO uniformly on compact subsetsof R+, where z is a solution of y’ = G(t, y) such that z(0) = y,, , and R+ is the set of t 2 0. Clearly for fixed t > 0, there exists a k sufficiently large so that
I @> I 3 Iyr(t> I - IyIm - z(t) I > y
(5)
where we have also used (4). Also since 1yk(t) 1 < r0 for t > 0 and k = 1, 2,..., it follows that
I z(t) I < y.
for
5>
0.
(6)
Now since g is in H(f), it follows that f is in H(g), and there exists a sequence (~~1, rk -+ + co as k + CO such that g(t + rk , x) -f (t, x) as k---f co uniformly for t in R and x in any compact subsetof R”. By use of Lemma 1 again, there exists a subsequenceof {T%},which we again denote by (Q}, such that #(t + Q) +$(t) as R+ co uniformly on compact subsetsof R; here 8 is a solution of (2) such that I p(t) I < B for t in R. If we define et,
Y) =f
(t9 Y + G(t)) -f
(6 F(t)),
it clearly follows that G(t + Tk , y) +F(t, y) as k + co, uniformly on compact subsetsof R x Rn. Clearly for each integer k, zk(t) = z(t + Q) is a solution of y’ = G(t + 7k , y), and by use of Lemma 1 again, there exists a subsequenceof {TV}, which we again denote by {Q}, such that zk(t) +9(t) as k + co uniformly on compact subsetsof R+; here y(t) is a solution of y’ = p(t, y). Now for fixed t > 0, there exists a R so large that
19(t) I z I +(t) I - I %(t) -7(t) I a *q where we have used (5) and the fact that 7k > 0 for K sufficiently large. Also from (6) and this samefact, it follows that Iy”(0) I < r, . But this implies y(t) + 0 as t + + co, which contradicts (7). This proves the theorem. COROLLARY 1. If f is a.p.u. (Rn), and x = 0 is a solution w.u.a.s., then it is u.a.s.
of (2) which is
ALMOST PROOF.
This corollary follows
PERIODIC
143
SOLUTIONS
immediately
from Theorem
1 by taking
B -0.
We observe here that Miller and Conley have given an example of a linear scalarsystem in which f is a.p.u. (P), the zero solution is asymptotically stable, but not uniformly stable; cf. [9]. 2. Let the hypotheses of Theorem I be saiisfied and in addition, suppose for each g in H(f) there exists a constant L(g) such that THEOREM
I ‘dc 4 - g(4 Y) f L(g) I A- -Y
I
for 1 x ) < B, 1y 1 < B, and 5 in R. Then each so&ion 1 p)(t) 1 < B for aZZ t in R is a.p.
g, of (2) such Ihat
PROOF. We first show that eachsolution ‘pof (2) satisfying the hypotheses of this theorem is separated(in the senseof Amerio [lo]) in the set S(B) of all x in Rn such that 1x I < B; i.e., for eachsuch v we show that there exists p(g)) > 0 such that for any solution +!J of (2) distinct from y such that / #(t)
I F(t) - (Ir(t>I 2 P(cp> for all
tER.
Since by Theorem 1 eachsuch 9 is u.a.s., each such ‘p is totally stable (i.e., stable under constantly acting disturbances) [3]. Supposesomesuch 9” = p1 is not separatedin S(B). Since ~‘1is u.a.s., there exists 6, > 0 such that if to is in R, then as 2+ co, I x(t, to ?x0) - dt) I - 0 provided 1x0 - q(to) / < 6,. But if v1 is not separatedin S(B), there exists a solution 9+ # p1 such that / pJt) 1< B for all t in R and 1yz(to) -- cpl(to) 1 < Sofor someto in R. Hence / cpz(t) - yl(t) / -+ 0 ast -+ co, and it follows also from the facts that v1 is u.a.s. and v1 # ~~ , that there exists a 6, > 0 such that 1q+(t) - VI(t) 1> 8, for t < to. Let {tk} be a sequencesuch that as K + 03, we have t, -+ - 00, f (t + t, , x) -f (t, cc), and Fj(t + tk) --+ a,+(t), j = 1,2; here the #j are solutions of (2), and the convergence is uniform on compact subsetsof R x R* and R, respectively (cf. Lemma 1). It follows easily that ( &(t) - $,(Q 2 6, for each t in R; we omit the details. Now pj(t + tk) is a solution of
x’ =f(c 4 + (f (t + h, 4 -f (C x)),
(4.1)
and #j is a solution of (2). Hence it follows from the total stability of #j that there exists a N(6,) > 0 such that for K > N(S,), we have for
t 20,
j=
1,2.
144
SEIFERT
For each such K and t > 0, it follows
easily that
and since also 1v,(t + tk) - v,(t + tk) ( + 0 as t -+ CO, we have for t sufficiently large that j #r(t) - #s(t) / < 6, , a contradiction. We conclude that yI is separated in S(B). We next show that if g is in H(f), then each solution # of x’ = g(t, x) Gw such that 1#(t) j < B f or all t in R is separated in S(B). To this end, we first observe that the set of such solutions of (2g) is necessarily finite. Let us denote the set of such solutions of (2) by {vj},j = 1,2,..., m. There exists a sequence {tk}, k = 1, 2,..., such that as k --f co, t, -+ + co, f(t + tk , X) -+g(t, x) (cf. (v)), and qj(t + tic) + #j(t) for j = 1, 2,..., m; here the convergence is uniform on compact subsets of R x Rn and R, respectively, and #j are solution of (2g). Since each p)j , j = I,2 ,..., m, is u.a.s., it follows that each #j , j = 1, 2,..., m, is a u.a.s. solution of (2g); cf. Lemma 4 in [7]. Now let # be a solution of (2g) such that 1z/(t) / < B for all t in R and such that 4 # #j , j = 1, 2,..., m. By using a suitable subsequence of {tk}, which we again denote by {tk}, it follows that #(t - tk) + Fdt) for some j, 1 < j < m, uniformly because g(t - t, , x) +f(t, x) as because there are no solutions of (2) the vi . However, there also exists using a suitable subsequence of (tk},
as
R+co
(4.2)
on compact subsets of R; this is true K -+ co uniformly on R x S(B), and with values contained in S(B) other than an integer i, 1 < i ,< m, such that by which we again denote by {tk}, we have
(bdt - tk> -+ 944
as
k-+co
(4.3)
uniformly on compact subsets of R. For if this were not true, there would exist integers p, 4, and r, p # 4, r fj, such that &,(t - tJ -+ ~)r(t) and &(t - tk) ---f y+(t) as k -+ co uniformly on compact subsets of R, where again a suitable subsequence, denoted by (tk}, is involved. But then 1#,( - tk) - z+,(- tk) ) -+ 0 as K --f co, and since tk -+ + cc as k + CO, and 9, # &, , it would follow that #, cannot be a.u.s. solution of (2g). From (4.2) and (4.3) it follows that / #( - tk) - Q&(- tk) + 0 as k -+ co, and since &. is a U.S. solution of (2g), we conclude that # = +t Yence the set of solutions with values in S(B) for all t in R consists of the fim set {I#~}, j = 1, 2,..., m. By using the same argument by which we proved ea I solution vi of (2) was separated in S(B), we find that each tij , j = 1, 2,..., 1, is separated in S(B) as a solution of (2g).
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We finally apply a result due to Amerio [lo] and the theorem is proved. We turn now to some results involving certain types of asymptotic stability in the large. LEMMA 3. Letfbe a.p.u. (Rn), and suppose v is a solution of (2) bounded on R and w.u.a.s.1. Then 9 is the only solution of (2) bounded on R. PROOF. Suppose Q!Iis another solution of (2) bounded on R. Since f is a.p.u. (Rn), there exists a sequence {tr}, t, ---f - co as k-t co, such that f (t + t, , X) -f (t, X) uniformly for t in R and x in any compact subset of R”; cf. (iv). Using Lemma 1, it follows that there exists a subsequenceof {tk) which we again denote by {tk} such that p)(t + tr) -+ q(t) and #(t + tk) - $ (t) as k + co uniformly on compact subsetsof R; here q and 4 are solutions of (2). Thus G(t) - p)(t) -+ 0 and $(t) - q(t) -+ 0 as t + + 00, and it follows that
$5(t) - $(t) --+ 0
as
t--t+00
(8)
Suppose there exists a to in R such that 1I - $(t,) 1 = l 0 > 0. It follows, since v is U.S.,that there exists a y0 > 0 such that for
I 40 - 6(t) I 2 ro
t < to 7
(9)
and also,by (8) that there exists tl in R such that
(10) Clearly for all k sufficiently large, we have t, + t, < to , and hence, using (9), I +(tl + tk) - v(t, + tk) I > ro. It follows that I J(h) - $(tl) I 3 y. which contradicts (10). Thus q(t) = $(t) for all t in R and our proof is complete. REMARK 7. It is easily seenthat the conclusion of Lemma 3 will hold if the hypothesis that f is a.p.u. (Rn) is replaced by an obviously weaker one; namely, that there exists a sequence{td, tk + - cc as k -+ 00, such that f (t + tk , x) -+f (t, x) as k -+ co, uniformly on compact subsetsof R x Rn.
LEMMA 4. Let the hypotheses of Lemma 3 be satisfied suppose that for each g in H( f ), the initial value problem for x’ =g(t, has a unique solution. Then solution bounded on R. PROOF.
f (t + c&21
t,
I-IO
for
x)
euch equation
and in addition,
GM (2g) there exists one and only one
Since g is in H(f), there exists a sequence {tk} such that , x) +g(t, x) as k -+ 00 uniformly for t in R and x in any compact
146
SEIFERT
subsetof Rn. It is no lossof generality to supposealsothat CJJ(~ + tJ -+ #r(t) as k -+ COuniformly on compact subsetsof R; hence t,+(t) is a solution of (2g) bounded on R. Let #a be another solution of (2g) bounded on R. Then since g(t - t, , x) +f(t, x) as k + cc uniformly for t in R and x in any compact subset of Rn, we have, by use of an appropriate subsequenceif necessary, which we again denote by {tk), that w
- tk) + W)
and
$z(t - tk) + f&(t)
as
k -+ co
uniformly on compact subsetsof R; here @jrand +a are bounded solutions of (2). By Lemma 3, ~$5~ = & = v, and it follows that ) #l(t - tk) - &(t - tk) / + 0
as
K + 0~)
(11)
uniformly for t in compact subsetsof R. We now show that h(t) - y&(t) -+ 0 as t--j + co. To this end, we assert that under the conditions of our lemma, the solution #r of (2g) is U.S.; for a proof of this, cf. Lemma 4 in [7]. Thus given E > 0, there exists a 8(e) > 0 as specified by the uniform stability of #r , and consequently alsoa k = K(c) such that I A(- tk) - Atk) I < %4; this last estimatefollows clearly from (11). It follows that I VW) - #z(t) I < e
for
t>
--k,
and therefore we have shown that a+&(t)- q&(t) + 0 as t --+ CO,and conclude that #r is a solution of (2g) which is w.u.a.s.1. But then, by Lemma 3 as applied to (2g) and +I , it follows that #r = #a , and the proof is complete. REMARK 8. It may be of interest to observe that the condition on the initial value problem for systems (2g) in the hypotheses of Lemma 3 is needed only to conclude, by Lemma 4 in [7], that the solution #r of (2g) is U.S.if its “prototype,” p is. Let f be a.p.u. (Rn). It is known that corresponding to such an f there exists a unique sequence{A,} of real numbers, the so-calledFourier exponents off. We call the set of all numbers of the form n,X, + nab, + *a*+ nkhk, where KZ~ ,..., nb are integers, the module off. THEOREM 3. Let f be a.p.u. (Rn) and suppose for each g in H(f) the initial value problems for (2g) have unique solutions. Let q~ be a solution of (2) which is w.u.a.s.1. and bounded on R. Then v is the only solutions of (2) which is bounded on R, is a.p., and its module is contained in the module off. PROOF.
The fact that y is a.p. follows immediately from Lemma 4 and
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a result due to Amerio [lo]. The fact that the module of 9 is contained in the module off follows from the fact that for each g in H(f), (2g) has exactly one solution bounded on R and that therefore the argument Favard [8] uses for linear systemscan be used here; we omit the details. It may be of interest to observe that this Theorem apparently does not follow from Lemma 3 and Theorem 2. In the first place we do not assumea Lipschitz condition here such as was required in Theorem 2, and we also do not get the information on the module of the a.p. solution from Theorem 2. COROLLARY 2. Let f be periodic in t with period w > 0, and suppose the initial value problems for (2) have unique solutions. Then if F is a solution of (2) which is w.u.a.s.1. and is bounded on R, it is periodic with period W. PROOF. If the initial value problems for (2) have unique solutions, it follows easily that the initial value problems for eachsystemx’ = f (t + h, x), where h is a constant, 0 < h < w, also have unique solutions. But in this case H(f) consists of all functions of the form f (t -\ h, x), 0 < h < O. Hence the hypothesesin Theorem 2 are satisfied. Since the module off is now (2~ k/w}, k = 0, & 1, f 2,..., the Fourier exponents of 91must be integer multiples of 2rr/w. From this it follows easily that C,Jis periodic with period W, and proof of the corollary is complete. This corollary sharpensSell’s Theorem 5 in [I] which assertsthe existence of a periodic solution of period RWfor someinteger k. The existence in this caseof a solution of period w hasalsobeen establishedby T. Yoshizawa using other methods. REMARK 9. Some apparently open questions suggest themselves. First, if f is a.p.u. (R”), can the hypothesis in Theorem 1 that all solutionsg, of (2) such that / y(t) 1< B for all t in R be w.u.a.s. be replacedby the condition that at least one such v has this stability property ? Also iff is periodic in t with period w the question may arise asto whether the existenceof a w.u.a.s. bounded solution implies the existenceof a periodic solution of period w. Sell’s Theorem 4 in [l] assertsonly the existence of a periodic solution of period KU, k an integer. The following example provides a negative answer to this last question. Define cos t sin t 44 = (- sin t cost)
consider the system:
E; = - (1 56= Ed1 - Cf22).
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SEIFERT
If E = (fl , ta), and f(f) = (- 4, , (a(1 formed by x = A(# into the system: x’ =Fyt,
Es2)), this
system
x)
is trans-
(12)
where qt,
x) = A’(t)
AT(t) x + A(t)f(AT(t)
x),
and
A-l(t) = AT(t) = (;g : - zb”,:) . By using the facts that f(E) = -f(f), A(t + CT) = - A(t), A’(t + CT) = - A’(t), and AT(t + n) = - AT(t), we observe that F(t + ?r, X) = F(t, X) for all (t, X) in R x R2. Now consider the 3-dimensional system x’ = F( t, x)
4 = Ax, 4,
(13)
where
1 - 2 I x 12,
I x I e +2;
here x = (xi , x2) and I x 1 = (xl2 + x2a)i/a. It follows that (13) is in t with period V, and that (- sin t, cos t, 0) and (sin t, - cos I, 0) tions of (13) of p eriod 27. It is easy to show that these are the only solutions of (13) and that they are u.a.s.; we omit the details. We also that any solution (x(t), xs(t)) of (13) such that
periodic are soluperiodic observe
(x(&J, x2(&)) = (u cos to , - a sin t, , c) where a and c are constants becomes unbounded as t + co, while any other solution tends to one of the two periodic solutions as t + CO. This example, and in particular, this last observation suggests the apparently open question: if each solution of a periodic system is bounded for t > 0 and if each solution bounded for all t is u.a.s., then is there among such bounded solutions a periodic solution with the same period as the system ? REMARK 10. It is clear that our hypotheses on the boundedness of solutions could have been weakened in the sense that any condition of the form “1 y(t) 1 < for all tin R” could be replaced by “a(t) in K, a compact subset of Rn, for all t in Rn”.
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REFERENCES 1. G. R. SELL. Periodic solutions and asymptotic stability. J. Diff. Eqs. 2 (1966), 143-157. 2. T. YOSHIZAWA. Stability theory by Liapunov’s second method. Math. Sot. Japan (1966). 3. J. MASSERA. Contributions to stability theory. Ann. Math. 64 (1956), 182-206. 4. R. K. MILLER. Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions. /. D$f. Eqs. 1 (1965), 337-345. 5. J. K. HALE. “Oscillations in Nonlinear Systems,” McGraw-Hill, New York, 1963. 6. E. KAMKE. Zur Theorie der Systeme gewijhnlicher Differentialgleichungen. II Acta Math. 58 (1932), 57-85. 7. G. SEIFERT. A condition for almost periodicity with some applications to functionaldifferential equations. J. Diff. Ep. 1 (1965), 393-408. 8. J. FAVARD. “Lecons sur les Fonctions Presque-Periodiques,” pp. 88-91. GauthierVillars, Paris, 1933. 9. C. C. CONLEY AND R. K. MILLER. “Asymptotic stability without uniform stability: almost periodic coefficients,” J. Diff. Eqs. 1 (1965), 333-336. 10. L. AMERIO. Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitati. Ann. Mat. Pura Appl. 34 (1955), 97-119.